- #1

- 14

- 1

## Homework Statement

Determine for what k f(x)=x

^{k}is an element of L

^{2}(0,1) vector space

k ∈ ℝ

## Homework Equations

## The Attempt at a Solution

[tex] \int_{0}^{1} x^{2k} dx = \frac{1-0^{2k+1}}{1+2k} = \sum_{n=0}^{\infty}{(-2k)^{n}} [/tex] (for k > -½)

This sum should converge for [tex]

\lim_{n \to +\infty}

{\frac{|(-2k)^{n+1}|}{|(-2k)^{n}|}} < 1

=

|-2k| < 1

[/tex]

Which gives me a radius of convergence for

[tex] - \frac{1}{2} < k < \frac{1}{2}

[/tex]

But just by examining it, the integral should exist for any k greater than negative one-half, what is wrong with my ratio test?

Last edited: